There are often several ways to answer a Math IIC question.
You can use trial and error, you can set up and solve an equation,
and, for some questions, you might be able to answer the question
quickly, intuitively, and elegantly, if you can just spot how. These
different approaches to answering questions vary in the amount of
time they take. Trial and error generally takes the longest, while
the elegant method of relying on an intuitive understanding of conceptual
knowledge takes the least amount of time.

Take, for example, the following problem:

Which
has a greater area, a square with sides measuring 4 cm, or a circle
with a radius of the same length?

The most obvious way to solve this problem is simply to
plug 4 into the formula for the area of a square and area of a circle.
Let’s do it: Area of a square = s2,
so the area of this square = 42 =
16. Area of a circle = πr2,
so the area of this circle must therefore be π42 =
16π. 16π is obviously bigger than 16, so the
circle must be bigger. That worked nicely. But a faster approach
would have been to draw a quick to-scale diagram with the square
and circle superimposed.

An even quicker way would have been to understand the
equations of area for a square and circle so well that it was just obvious that
the circle was bigger, since the equation for the circle will square
the 4 and multiply it by whereas
the equation for the square will only square the 4.

While you may not be able to become a math whiz and just know the
answer, you can learn to look for a quicker route, such as choosing
to draw a diagram instead of working out the equation. And, as with
the example above, a quicker route is not necessarily a less accurate
one. Making such choices comes down to practice, being aware that
those other routes are out there, and basic mathematical ability.

The value of time-saving strategies is obvious: less time
spent on some questions allows you to devote more time to difficult
problems. It is this issue of time that separates the students who
do terrifically on the test and those who merely do well. Whether
or not the ability to find accurate shortcuts is an actual measure
of mathematical prowess is not for us to say, but the ability to
find those shortcuts absolutely matters on this test.

Shortcuts Are Really Math Intuition

Now that you know all about shortcuts, you should use
them wisely. Don’t go into every question searching for a shortcut;
it might end up taking longer than the normal route. Instead of
seeking out math shortcuts, you should simply be mindful of the
possibility that one might exist. If you go into each question knowing
there could be a shortcut and keep your mind open as you think about
the question, you will find the shortcuts you need.

To some extent, with practice you can teach yourself to
recognize when a question might contain a shortcut. For example,
simply from the problem above, you know that there will probably
be a shortcut for questions that give you the dimensions of two
shapes and ask you to compare them: you can just draw a diagram.
A frantic test-taker might see the information given and then seize
on the simplest route and work out the equations. But with some
calm and perspective you can see that drawing a diagram is the best
idea.

The fact that we advocate using shortcuts doesn’t mean
you shouldn’t focus on learning how to work out problems. In fact,
we can guarantee that you’re not going to find a shortcut for a
problem unless you know how to work it out the
long way. After all, a shortcut just uses your knowledge to find
a faster way to answer the question. When we use the term math
shortcut, we are really referring to your math intuition.